Random walks and bisections in random circulant graphs

Authors:
Bernard Mans;Igor E. Shparlinski
Affiliations:
Department of Computing, Macquarie University, Sydney, NSW, Australia;Department of Computing, Macquarie University, Sydney, NSW, Australia
Venue:
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
Year:
2012

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Abstract

Using number theoretical tools, we prove two main results for random r-regular circulant graphs with n vertices, when n is sufficiently large and r is fixed. First, for any fixed ε0, prime n and L≥n1/r (logn)1+1/r+ε, walks of length at most L terminate at every vertex with asymptotically the same probability. Second, for any n, there is a polynomial time algorithm to find a vertex bisector and an edge bisector, both of size less than n1−1/r+o(1). As circulant graphs are popular network topologies in distributed computing, we show that our results can be exploited for various information dissemination schemes. In particular, we provide lower bounds on the number of rounds required by any gossiping algorithms for any n. This settles an open question in an earlier work of the authors (2004) and shows that the generic gossiping algorithms of that work are nearly optimal.